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Theorem sge0tsms 44611
Description: Σ^ applied to a nonnegative function (its meaningful domain) is the same as the infinite group sum (that's always convergent, in this case). (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
sge0tsms.g 𝐺 = (ℝ*𝑠s (0[,]+∞))
sge0tsms.x (𝜑𝑋𝑉)
sge0tsms.f (𝜑𝐹:𝑋⟶(0[,]+∞))
Assertion
Ref Expression
sge0tsms (𝜑 → (Σ^𝐹) ∈ (𝐺 tsums 𝐹))

Proof of Theorem sge0tsms
Dummy variables 𝑠 𝑡 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2736 . . . 4 sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < )
21a1i 11 . . 3 (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ))
3 xrltso 13060 . . . . . 6 < Or ℝ*
43supex 9399 . . . . 5 sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) ∈ V
54a1i 11 . . . 4 (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) ∈ V)
6 elsng 4600 . . . 4 (sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) ∈ V → (sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) ∈ {sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < )} ↔ sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < )))
75, 6syl 17 . . 3 (𝜑 → (sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) ∈ {sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < )} ↔ sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < )))
82, 7mpbird 256 . 2 (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) ∈ {sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < )})
9 sge0tsms.x . . . . . . 7 (𝜑𝑋𝑉)
109adantr 481 . . . . . 6 ((𝜑 ∧ +∞ ∈ ran 𝐹) → 𝑋𝑉)
11 sge0tsms.f . . . . . . 7 (𝜑𝐹:𝑋⟶(0[,]+∞))
1211adantr 481 . . . . . 6 ((𝜑 ∧ +∞ ∈ ran 𝐹) → 𝐹:𝑋⟶(0[,]+∞))
13 simpr 485 . . . . . 6 ((𝜑 ∧ +∞ ∈ ran 𝐹) → +∞ ∈ ran 𝐹)
1410, 12, 13sge0pnfval 44604 . . . . 5 ((𝜑 ∧ +∞ ∈ ran 𝐹) → (Σ^𝐹) = +∞)
1511ffnd 6669 . . . . . . . . 9 (𝜑𝐹 Fn 𝑋)
1615adantr 481 . . . . . . . 8 ((𝜑 ∧ +∞ ∈ ran 𝐹) → 𝐹 Fn 𝑋)
17 fvelrnb 6903 . . . . . . . 8 (𝐹 Fn 𝑋 → (+∞ ∈ ran 𝐹 ↔ ∃𝑦𝑋 (𝐹𝑦) = +∞))
1816, 17syl 17 . . . . . . 7 ((𝜑 ∧ +∞ ∈ ran 𝐹) → (+∞ ∈ ran 𝐹 ↔ ∃𝑦𝑋 (𝐹𝑦) = +∞))
1913, 18mpbid 231 . . . . . 6 ((𝜑 ∧ +∞ ∈ ran 𝐹) → ∃𝑦𝑋 (𝐹𝑦) = +∞)
20 iccssxr 13347 . . . . . . . . . . . . . 14 (0[,]+∞) ⊆ ℝ*
21 sge0tsms.g . . . . . . . . . . . . . . 15 𝐺 = (ℝ*𝑠s (0[,]+∞))
22 simpr 485 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ∈ (𝒫 𝑋 ∩ Fin))
2311adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝐹:𝑋⟶(0[,]+∞))
24 elinel1 4155 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥 ∈ 𝒫 𝑋)
25 elpwi 4567 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ 𝒫 𝑋𝑥𝑋)
2624, 25syl 17 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥𝑋)
2726adantl 482 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥𝑋)
28 fssres 6708 . . . . . . . . . . . . . . . 16 ((𝐹:𝑋⟶(0[,]+∞) ∧ 𝑥𝑋) → (𝐹𝑥):𝑥⟶(0[,]+∞))
2923, 27, 28syl2anc 584 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐹𝑥):𝑥⟶(0[,]+∞))
30 elinel2 4156 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥 ∈ Fin)
3130adantl 482 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ∈ Fin)
32 0red 11158 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 0 ∈ ℝ)
3329, 31, 32fdmfifsupp 9315 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐹𝑥) finSupp 0)
3421, 22, 29, 33gsumge0cl 44602 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐺 Σg (𝐹𝑥)) ∈ (0[,]+∞))
3520, 34sselid 3942 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐺 Σg (𝐹𝑥)) ∈ ℝ*)
3635ralrimiva 3143 . . . . . . . . . . . 12 (𝜑 → ∀𝑥 ∈ (𝒫 𝑋 ∩ Fin)(𝐺 Σg (𝐹𝑥)) ∈ ℝ*)
37363ad2ant1 1133 . . . . . . . . . . 11 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → ∀𝑥 ∈ (𝒫 𝑋 ∩ Fin)(𝐺 Σg (𝐹𝑥)) ∈ ℝ*)
38 eqid 2736 . . . . . . . . . . . 12 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))) = (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥)))
3938rnmptss 7070 . . . . . . . . . . 11 (∀𝑥 ∈ (𝒫 𝑋 ∩ Fin)(𝐺 Σg (𝐹𝑥)) ∈ ℝ* → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))) ⊆ ℝ*)
4037, 39syl 17 . . . . . . . . . 10 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))) ⊆ ℝ*)
41 snelpwi 5400 . . . . . . . . . . . . . 14 (𝑦𝑋 → {𝑦} ∈ 𝒫 𝑋)
42 snfi 8988 . . . . . . . . . . . . . . 15 {𝑦} ∈ Fin
4342a1i 11 . . . . . . . . . . . . . 14 (𝑦𝑋 → {𝑦} ∈ Fin)
4441, 43elind 4154 . . . . . . . . . . . . 13 (𝑦𝑋 → {𝑦} ∈ (𝒫 𝑋 ∩ Fin))
45443ad2ant2 1134 . . . . . . . . . . . 12 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → {𝑦} ∈ (𝒫 𝑋 ∩ Fin))
4611adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑦𝑋) → 𝐹:𝑋⟶(0[,]+∞))
47 snssi 4768 . . . . . . . . . . . . . . . . . . 19 (𝑦𝑋 → {𝑦} ⊆ 𝑋)
4847adantl 482 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑦𝑋) → {𝑦} ⊆ 𝑋)
4946, 48fssresd 6709 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦𝑋) → (𝐹 ↾ {𝑦}):{𝑦}⟶(0[,]+∞))
5049feqmptd 6910 . . . . . . . . . . . . . . . 16 ((𝜑𝑦𝑋) → (𝐹 ↾ {𝑦}) = (𝑥 ∈ {𝑦} ↦ ((𝐹 ↾ {𝑦})‘𝑥)))
51 fvres 6861 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ {𝑦} → ((𝐹 ↾ {𝑦})‘𝑥) = (𝐹𝑥))
5251mpteq2ia 5208 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ {𝑦} ↦ ((𝐹 ↾ {𝑦})‘𝑥)) = (𝑥 ∈ {𝑦} ↦ (𝐹𝑥))
5352a1i 11 . . . . . . . . . . . . . . . 16 ((𝜑𝑦𝑋) → (𝑥 ∈ {𝑦} ↦ ((𝐹 ↾ {𝑦})‘𝑥)) = (𝑥 ∈ {𝑦} ↦ (𝐹𝑥)))
5450, 53eqtrd 2776 . . . . . . . . . . . . . . 15 ((𝜑𝑦𝑋) → (𝐹 ↾ {𝑦}) = (𝑥 ∈ {𝑦} ↦ (𝐹𝑥)))
5554oveq2d 7373 . . . . . . . . . . . . . 14 ((𝜑𝑦𝑋) → (𝐺 Σg (𝐹 ↾ {𝑦})) = (𝐺 Σg (𝑥 ∈ {𝑦} ↦ (𝐹𝑥))))
56553adant3 1132 . . . . . . . . . . . . 13 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → (𝐺 Σg (𝐹 ↾ {𝑦})) = (𝐺 Σg (𝑥 ∈ {𝑦} ↦ (𝐹𝑥))))
57 xrge0cmn 20839 . . . . . . . . . . . . . . . . 17 (ℝ*𝑠s (0[,]+∞)) ∈ CMnd
5821, 57eqeltri 2834 . . . . . . . . . . . . . . . 16 𝐺 ∈ CMnd
59 cmnmnd 19579 . . . . . . . . . . . . . . . 16 (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
6058, 59ax-mp 5 . . . . . . . . . . . . . . 15 𝐺 ∈ Mnd
6160a1i 11 . . . . . . . . . . . . . 14 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → 𝐺 ∈ Mnd)
62 simp2 1137 . . . . . . . . . . . . . 14 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → 𝑦𝑋)
6311ffvelcdmda 7035 . . . . . . . . . . . . . . 15 ((𝜑𝑦𝑋) → (𝐹𝑦) ∈ (0[,]+∞))
64633adant3 1132 . . . . . . . . . . . . . 14 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → (𝐹𝑦) ∈ (0[,]+∞))
65 df-ss 3927 . . . . . . . . . . . . . . . . . 18 ((0[,]+∞) ⊆ ℝ* ↔ ((0[,]+∞) ∩ ℝ*) = (0[,]+∞))
6620, 65mpbi 229 . . . . . . . . . . . . . . . . 17 ((0[,]+∞) ∩ ℝ*) = (0[,]+∞)
6766eqcomi 2745 . . . . . . . . . . . . . . . 16 (0[,]+∞) = ((0[,]+∞) ∩ ℝ*)
68 ovex 7390 . . . . . . . . . . . . . . . . 17 (0[,]+∞) ∈ V
69 xrsbas 20813 . . . . . . . . . . . . . . . . . 18 * = (Base‘ℝ*𝑠)
7021, 69ressbas 17118 . . . . . . . . . . . . . . . . 17 ((0[,]+∞) ∈ V → ((0[,]+∞) ∩ ℝ*) = (Base‘𝐺))
7168, 70ax-mp 5 . . . . . . . . . . . . . . . 16 ((0[,]+∞) ∩ ℝ*) = (Base‘𝐺)
7267, 71eqtri 2764 . . . . . . . . . . . . . . 15 (0[,]+∞) = (Base‘𝐺)
73 fveq2 6842 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
7472, 73gsumsn 19731 . . . . . . . . . . . . . 14 ((𝐺 ∈ Mnd ∧ 𝑦𝑋 ∧ (𝐹𝑦) ∈ (0[,]+∞)) → (𝐺 Σg (𝑥 ∈ {𝑦} ↦ (𝐹𝑥))) = (𝐹𝑦))
7561, 62, 64, 74syl3anc 1371 . . . . . . . . . . . . 13 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → (𝐺 Σg (𝑥 ∈ {𝑦} ↦ (𝐹𝑥))) = (𝐹𝑦))
76 simp3 1138 . . . . . . . . . . . . 13 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → (𝐹𝑦) = +∞)
7756, 75, 763eqtrrd 2781 . . . . . . . . . . . 12 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → +∞ = (𝐺 Σg (𝐹 ↾ {𝑦})))
78 reseq2 5932 . . . . . . . . . . . . . 14 (𝑥 = {𝑦} → (𝐹𝑥) = (𝐹 ↾ {𝑦}))
7978oveq2d 7373 . . . . . . . . . . . . 13 (𝑥 = {𝑦} → (𝐺 Σg (𝐹𝑥)) = (𝐺 Σg (𝐹 ↾ {𝑦})))
8079rspceeqv 3595 . . . . . . . . . . . 12 (({𝑦} ∈ (𝒫 𝑋 ∩ Fin) ∧ +∞ = (𝐺 Σg (𝐹 ↾ {𝑦}))) → ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)+∞ = (𝐺 Σg (𝐹𝑥)))
8145, 77, 80syl2anc 584 . . . . . . . . . . 11 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)+∞ = (𝐺 Σg (𝐹𝑥)))
82 pnfxr 11209 . . . . . . . . . . . . 13 +∞ ∈ ℝ*
8382a1i 11 . . . . . . . . . . . 12 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → +∞ ∈ ℝ*)
8438elrnmpt 5911 . . . . . . . . . . . 12 (+∞ ∈ ℝ* → (+∞ ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))) ↔ ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)+∞ = (𝐺 Σg (𝐹𝑥))))
8583, 84syl 17 . . . . . . . . . . 11 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → (+∞ ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))) ↔ ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)+∞ = (𝐺 Σg (𝐹𝑥))))
8681, 85mpbird 256 . . . . . . . . . 10 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → +∞ ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))))
87 supxrpnf 13237 . . . . . . . . . 10 ((ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))) ⊆ ℝ* ∧ +∞ ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥)))) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) = +∞)
8840, 86, 87syl2anc 584 . . . . . . . . 9 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) = +∞)
89883exp 1119 . . . . . . . 8 (𝜑 → (𝑦𝑋 → ((𝐹𝑦) = +∞ → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) = +∞)))
9089adantr 481 . . . . . . 7 ((𝜑 ∧ +∞ ∈ ran 𝐹) → (𝑦𝑋 → ((𝐹𝑦) = +∞ → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) = +∞)))
9190rexlimdv 3150 . . . . . 6 ((𝜑 ∧ +∞ ∈ ran 𝐹) → (∃𝑦𝑋 (𝐹𝑦) = +∞ → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) = +∞))
9219, 91mpd 15 . . . . 5 ((𝜑 ∧ +∞ ∈ ran 𝐹) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) = +∞)
9314, 92eqtr4d 2779 . . . 4 ((𝜑 ∧ +∞ ∈ ran 𝐹) → (Σ^𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ))
949adantr 481 . . . . . 6 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → 𝑋𝑉)
9511adantr 481 . . . . . . 7 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → 𝐹:𝑋⟶(0[,]+∞))
96 simpr 485 . . . . . . 7 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → ¬ +∞ ∈ ran 𝐹)
9795, 96fge0iccico 44601 . . . . . 6 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → 𝐹:𝑋⟶(0[,)+∞))
9894, 97sge0reval 44603 . . . . 5 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ*, < ))
9923, 27feqresmpt 6911 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐹𝑥) = (𝑦𝑥 ↦ (𝐹𝑦)))
10099adantlr 713 . . . . . . . . . 10 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐹𝑥) = (𝑦𝑥 ↦ (𝐹𝑦)))
101100oveq2d 7373 . . . . . . . . 9 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐺 Σg (𝐹𝑥)) = (𝐺 Σg (𝑦𝑥 ↦ (𝐹𝑦))))
10221fveq2i 6845 . . . . . . . . . . 11 (+g𝐺) = (+g‘(ℝ*𝑠s (0[,]+∞)))
103 eqid 2736 . . . . . . . . . . . . . 14 (ℝ*𝑠s (0[,]+∞)) = (ℝ*𝑠s (0[,]+∞))
104 xrsadd 20814 . . . . . . . . . . . . . 14 +𝑒 = (+g‘ℝ*𝑠)
105103, 104ressplusg 17171 . . . . . . . . . . . . 13 ((0[,]+∞) ∈ V → +𝑒 = (+g‘(ℝ*𝑠s (0[,]+∞))))
10668, 105ax-mp 5 . . . . . . . . . . . 12 +𝑒 = (+g‘(ℝ*𝑠s (0[,]+∞)))
107106eqcomi 2745 . . . . . . . . . . 11 (+g‘(ℝ*𝑠s (0[,]+∞))) = +𝑒
108102, 107eqtr2i 2765 . . . . . . . . . 10 +𝑒 = (+g𝐺)
10921oveq1i 7367 . . . . . . . . . . 11 (𝐺s (0[,)+∞)) = ((ℝ*𝑠s (0[,]+∞)) ↾s (0[,)+∞))
110 icossicc 13353 . . . . . . . . . . . . 13 (0[,)+∞) ⊆ (0[,]+∞)
11168, 110pm3.2i 471 . . . . . . . . . . . 12 ((0[,]+∞) ∈ V ∧ (0[,)+∞) ⊆ (0[,]+∞))
112 ressabs 17130 . . . . . . . . . . . 12 (((0[,]+∞) ∈ V ∧ (0[,)+∞) ⊆ (0[,]+∞)) → ((ℝ*𝑠s (0[,]+∞)) ↾s (0[,)+∞)) = (ℝ*𝑠s (0[,)+∞)))
113111, 112ax-mp 5 . . . . . . . . . . 11 ((ℝ*𝑠s (0[,]+∞)) ↾s (0[,)+∞)) = (ℝ*𝑠s (0[,)+∞))
114109, 113eqtr2i 2765 . . . . . . . . . 10 (ℝ*𝑠s (0[,)+∞)) = (𝐺s (0[,)+∞))
11558elexi 3464 . . . . . . . . . . 11 𝐺 ∈ V
116115a1i 11 . . . . . . . . . 10 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝐺 ∈ V)
117 simpr 485 . . . . . . . . . 10 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ∈ (𝒫 𝑋 ∩ Fin))
118110a1i 11 . . . . . . . . . 10 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (0[,)+∞) ⊆ (0[,]+∞))
119 0xr 11202 . . . . . . . . . . . . 13 0 ∈ ℝ*
120119a1i 11 . . . . . . . . . . . 12 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → 0 ∈ ℝ*)
12182a1i 11 . . . . . . . . . . . 12 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → +∞ ∈ ℝ*)
12295ad2antrr 724 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → 𝐹:𝑋⟶(0[,]+∞))
12326sselda 3944 . . . . . . . . . . . . . . 15 ((𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑦𝑥) → 𝑦𝑋)
124123adantll 712 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → 𝑦𝑋)
125122, 124ffvelcdmd 7036 . . . . . . . . . . . . 13 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ (0[,]+∞))
12620, 125sselid 3942 . . . . . . . . . . . 12 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ ℝ*)
127 iccgelb 13320 . . . . . . . . . . . . 13 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ (𝐹𝑦) ∈ (0[,]+∞)) → 0 ≤ (𝐹𝑦))
128120, 121, 125, 127syl3anc 1371 . . . . . . . . . . . 12 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → 0 ≤ (𝐹𝑦))
129 id 22 . . . . . . . . . . . . . . . . . . . 20 ((𝐹𝑦) = +∞ → (𝐹𝑦) = +∞)
130129eqcomd 2742 . . . . . . . . . . . . . . . . . . 19 ((𝐹𝑦) = +∞ → +∞ = (𝐹𝑦))
131130adantl 482 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) ∧ (𝐹𝑦) = +∞) → +∞ = (𝐹𝑦))
13211ffund 6672 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → Fun 𝐹)
133132ad2antrr 724 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → Fun 𝐹)
13422, 123sylan 580 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → 𝑦𝑋)
13511fdmd 6679 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → dom 𝐹 = 𝑋)
136135eqcomd 2742 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝑋 = dom 𝐹)
137136ad2antrr 724 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → 𝑋 = dom 𝐹)
138134, 137eleqtrd 2840 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → 𝑦 ∈ dom 𝐹)
139 fvelrn 7027 . . . . . . . . . . . . . . . . . . . 20 ((Fun 𝐹𝑦 ∈ dom 𝐹) → (𝐹𝑦) ∈ ran 𝐹)
140133, 138, 139syl2anc 584 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ ran 𝐹)
141140adantr 481 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) ∧ (𝐹𝑦) = +∞) → (𝐹𝑦) ∈ ran 𝐹)
142131, 141eqeltrd 2838 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) ∧ (𝐹𝑦) = +∞) → +∞ ∈ ran 𝐹)
143142adantl3r 748 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) ∧ (𝐹𝑦) = +∞) → +∞ ∈ ran 𝐹)
14496ad3antrrr 728 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) ∧ (𝐹𝑦) = +∞) → ¬ +∞ ∈ ran 𝐹)
145143, 144pm2.65da 815 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → ¬ (𝐹𝑦) = +∞)
146145neqned 2950 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ≠ +∞)
147 ge0xrre 43759 . . . . . . . . . . . . . 14 (((𝐹𝑦) ∈ (0[,]+∞) ∧ (𝐹𝑦) ≠ +∞) → (𝐹𝑦) ∈ ℝ)
148125, 146, 147syl2anc 584 . . . . . . . . . . . . 13 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ ℝ)
149148ltpnfd 13042 . . . . . . . . . . . 12 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) < +∞)
150120, 121, 126, 128, 149elicod 13314 . . . . . . . . . . 11 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ (0[,)+∞))
151 eqid 2736 . . . . . . . . . . 11 (𝑦𝑥 ↦ (𝐹𝑦)) = (𝑦𝑥 ↦ (𝐹𝑦))
152150, 151fmptd 7062 . . . . . . . . . 10 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝑦𝑥 ↦ (𝐹𝑦)):𝑥⟶(0[,)+∞))
153 0e0icopnf 13375 . . . . . . . . . . 11 0 ∈ (0[,)+∞)
154153a1i 11 . . . . . . . . . 10 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 0 ∈ (0[,)+∞))
155 eliccxr 13352 . . . . . . . . . . . 12 (𝑦 ∈ (0[,]+∞) → 𝑦 ∈ ℝ*)
156 xaddid2 13161 . . . . . . . . . . . . 13 (𝑦 ∈ ℝ* → (0 +𝑒 𝑦) = 𝑦)
157 xaddid1 13160 . . . . . . . . . . . . 13 (𝑦 ∈ ℝ* → (𝑦 +𝑒 0) = 𝑦)
158156, 157jca 512 . . . . . . . . . . . 12 (𝑦 ∈ ℝ* → ((0 +𝑒 𝑦) = 𝑦 ∧ (𝑦 +𝑒 0) = 𝑦))
159155, 158syl 17 . . . . . . . . . . 11 (𝑦 ∈ (0[,]+∞) → ((0 +𝑒 𝑦) = 𝑦 ∧ (𝑦 +𝑒 0) = 𝑦))
160159adantl 482 . . . . . . . . . 10 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ (0[,]+∞)) → ((0 +𝑒 𝑦) = 𝑦 ∧ (𝑦 +𝑒 0) = 𝑦))
16172, 108, 114, 116, 117, 118, 152, 154, 160gsumress 18537 . . . . . . . . 9 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐺 Σg (𝑦𝑥 ↦ (𝐹𝑦))) = ((ℝ*𝑠s (0[,)+∞)) Σg (𝑦𝑥 ↦ (𝐹𝑦))))
162 rege0subm 20853 . . . . . . . . . . . . 13 (0[,)+∞) ∈ (SubMnd‘ℂfld)
163162a1i 11 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (0[,)+∞) ∈ (SubMnd‘ℂfld))
164 eqid 2736 . . . . . . . . . . . 12 (ℂflds (0[,)+∞)) = (ℂflds (0[,)+∞))
165117, 163, 152, 164gsumsubm 18645 . . . . . . . . . . 11 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (ℂfld Σg (𝑦𝑥 ↦ (𝐹𝑦))) = ((ℂflds (0[,)+∞)) Σg (𝑦𝑥 ↦ (𝐹𝑦))))
166 eqidd 2737 . . . . . . . . . . 11 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → ((ℂflds (0[,)+∞)) Σg (𝑦𝑥 ↦ (𝐹𝑦))) = ((ℂflds (0[,)+∞)) Σg (𝑦𝑥 ↦ (𝐹𝑦))))
167 vex 3449 . . . . . . . . . . . . . 14 𝑥 ∈ V
168167mptex 7173 . . . . . . . . . . . . 13 (𝑦𝑥 ↦ (𝐹𝑦)) ∈ V
169168a1i 11 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝑦𝑥 ↦ (𝐹𝑦)) ∈ V)
170 ovexd 7392 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (ℂflds (0[,)+∞)) ∈ V)
171 ovexd 7392 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (ℝ*𝑠s (0[,)+∞)) ∈ V)
172 rge0ssre 13373 . . . . . . . . . . . . . . . . 17 (0[,)+∞) ⊆ ℝ
173 ax-resscn 11108 . . . . . . . . . . . . . . . . 17 ℝ ⊆ ℂ
174172, 173sstri 3953 . . . . . . . . . . . . . . . 16 (0[,)+∞) ⊆ ℂ
175 cnfldbas 20800 . . . . . . . . . . . . . . . . 17 ℂ = (Base‘ℂfld)
176164, 175ressbas2 17120 . . . . . . . . . . . . . . . 16 ((0[,)+∞) ⊆ ℂ → (0[,)+∞) = (Base‘(ℂflds (0[,)+∞))))
177174, 176ax-mp 5 . . . . . . . . . . . . . . 15 (0[,)+∞) = (Base‘(ℂflds (0[,)+∞)))
178177eqcomi 2745 . . . . . . . . . . . . . 14 (Base‘(ℂflds (0[,)+∞))) = (0[,)+∞)
179110, 20sstri 3953 . . . . . . . . . . . . . . 15 (0[,)+∞) ⊆ ℝ*
180 eqid 2736 . . . . . . . . . . . . . . . 16 (ℝ*𝑠s (0[,)+∞)) = (ℝ*𝑠s (0[,)+∞))
181180, 69ressbas2 17120 . . . . . . . . . . . . . . 15 ((0[,)+∞) ⊆ ℝ* → (0[,)+∞) = (Base‘(ℝ*𝑠s (0[,)+∞))))
182179, 181ax-mp 5 . . . . . . . . . . . . . 14 (0[,)+∞) = (Base‘(ℝ*𝑠s (0[,)+∞)))
183178, 182eqtri 2764 . . . . . . . . . . . . 13 (Base‘(ℂflds (0[,)+∞))) = (Base‘(ℝ*𝑠s (0[,)+∞)))
184183a1i 11 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (Base‘(ℂflds (0[,)+∞))) = (Base‘(ℝ*𝑠s (0[,)+∞))))
185 rge0srg 20868 . . . . . . . . . . . . . . 15 (ℂflds (0[,)+∞)) ∈ SRing
186185a1i 11 . . . . . . . . . . . . . 14 ((𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂflds (0[,)+∞)))) → (ℂflds (0[,)+∞)) ∈ SRing)
187 simpl 483 . . . . . . . . . . . . . 14 ((𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂflds (0[,)+∞)))) → 𝑠 ∈ (Base‘(ℂflds (0[,)+∞))))
188 simpr 485 . . . . . . . . . . . . . 14 ((𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂflds (0[,)+∞)))) → 𝑡 ∈ (Base‘(ℂflds (0[,)+∞))))
189 eqid 2736 . . . . . . . . . . . . . . 15 (Base‘(ℂflds (0[,)+∞))) = (Base‘(ℂflds (0[,)+∞)))
190 eqid 2736 . . . . . . . . . . . . . . 15 (+g‘(ℂflds (0[,)+∞))) = (+g‘(ℂflds (0[,)+∞)))
191189, 190srgacl 19936 . . . . . . . . . . . . . 14 (((ℂflds (0[,)+∞)) ∈ SRing ∧ 𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂflds (0[,)+∞)))) → (𝑠(+g‘(ℂflds (0[,)+∞)))𝑡) ∈ (Base‘(ℂflds (0[,)+∞))))
192186, 187, 188, 191syl3anc 1371 . . . . . . . . . . . . 13 ((𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂflds (0[,)+∞)))) → (𝑠(+g‘(ℂflds (0[,)+∞)))𝑡) ∈ (Base‘(ℂflds (0[,)+∞))))
193192adantl 482 . . . . . . . . . . . 12 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ (𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂflds (0[,)+∞))))) → (𝑠(+g‘(ℂflds (0[,)+∞)))𝑡) ∈ (Base‘(ℂflds (0[,)+∞))))
194172a1i 11 . . . . . . . . . . . . . . . 16 (𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) → (0[,)+∞) ⊆ ℝ)
195 id 22 . . . . . . . . . . . . . . . . 17 (𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) → 𝑠 ∈ (Base‘(ℂflds (0[,)+∞))))
196195, 178eleqtrdi 2848 . . . . . . . . . . . . . . . 16 (𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) → 𝑠 ∈ (0[,)+∞))
197194, 196sseldd 3945 . . . . . . . . . . . . . . 15 (𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) → 𝑠 ∈ ℝ)
198197adantr 481 . . . . . . . . . . . . . 14 ((𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂflds (0[,)+∞)))) → 𝑠 ∈ ℝ)
199172a1i 11 . . . . . . . . . . . . . . . 16 (𝑡 ∈ (Base‘(ℂflds (0[,)+∞))) → (0[,)+∞) ⊆ ℝ)
200 id 22 . . . . . . . . . . . . . . . . 17 (𝑡 ∈ (Base‘(ℂflds (0[,)+∞))) → 𝑡 ∈ (Base‘(ℂflds (0[,)+∞))))
201200, 178eleqtrdi 2848 . . . . . . . . . . . . . . . 16 (𝑡 ∈ (Base‘(ℂflds (0[,)+∞))) → 𝑡 ∈ (0[,)+∞))
202199, 201sseldd 3945 . . . . . . . . . . . . . . 15 (𝑡 ∈ (Base‘(ℂflds (0[,)+∞))) → 𝑡 ∈ ℝ)
203202adantl 482 . . . . . . . . . . . . . 14 ((𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂflds (0[,)+∞)))) → 𝑡 ∈ ℝ)
204 rexadd 13151 . . . . . . . . . . . . . . . 16 ((𝑠 ∈ ℝ ∧ 𝑡 ∈ ℝ) → (𝑠 +𝑒 𝑡) = (𝑠 + 𝑡))
205204eqcomd 2742 . . . . . . . . . . . . . . 15 ((𝑠 ∈ ℝ ∧ 𝑡 ∈ ℝ) → (𝑠 + 𝑡) = (𝑠 +𝑒 𝑡))
206162elexi 3464 . . . . . . . . . . . . . . . . . . . 20 (0[,)+∞) ∈ V
207 cnfldadd 20801 . . . . . . . . . . . . . . . . . . . . 21 + = (+g‘ℂfld)
208164, 207ressplusg 17171 . . . . . . . . . . . . . . . . . . . 20 ((0[,)+∞) ∈ V → + = (+g‘(ℂflds (0[,)+∞))))
209206, 208ax-mp 5 . . . . . . . . . . . . . . . . . . 19 + = (+g‘(ℂflds (0[,)+∞)))
210209, 207eqtr3i 2766 . . . . . . . . . . . . . . . . . 18 (+g‘(ℂflds (0[,)+∞))) = (+g‘ℂfld)
211210, 207eqtr4i 2767 . . . . . . . . . . . . . . . . 17 (+g‘(ℂflds (0[,)+∞))) = +
212211oveqi 7370 . . . . . . . . . . . . . . . 16 (𝑠(+g‘(ℂflds (0[,)+∞)))𝑡) = (𝑠 + 𝑡)
213212a1i 11 . . . . . . . . . . . . . . 15 ((𝑠 ∈ ℝ ∧ 𝑡 ∈ ℝ) → (𝑠(+g‘(ℂflds (0[,)+∞)))𝑡) = (𝑠 + 𝑡))
214180, 104ressplusg 17171 . . . . . . . . . . . . . . . . . . 19 ((0[,)+∞) ∈ V → +𝑒 = (+g‘(ℝ*𝑠s (0[,)+∞))))
215206, 214ax-mp 5 . . . . . . . . . . . . . . . . . 18 +𝑒 = (+g‘(ℝ*𝑠s (0[,)+∞)))
216215eqcomi 2745 . . . . . . . . . . . . . . . . 17 (+g‘(ℝ*𝑠s (0[,)+∞))) = +𝑒
217216oveqi 7370 . . . . . . . . . . . . . . . 16 (𝑠(+g‘(ℝ*𝑠s (0[,)+∞)))𝑡) = (𝑠 +𝑒 𝑡)
218217a1i 11 . . . . . . . . . . . . . . 15 ((𝑠 ∈ ℝ ∧ 𝑡 ∈ ℝ) → (𝑠(+g‘(ℝ*𝑠s (0[,)+∞)))𝑡) = (𝑠 +𝑒 𝑡))
219205, 213, 2183eqtr4d 2786 . . . . . . . . . . . . . 14 ((𝑠 ∈ ℝ ∧ 𝑡 ∈ ℝ) → (𝑠(+g‘(ℂflds (0[,)+∞)))𝑡) = (𝑠(+g‘(ℝ*𝑠s (0[,)+∞)))𝑡))
220198, 203, 219syl2anc 584 . . . . . . . . . . . . 13 ((𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂflds (0[,)+∞)))) → (𝑠(+g‘(ℂflds (0[,)+∞)))𝑡) = (𝑠(+g‘(ℝ*𝑠s (0[,)+∞)))𝑡))
221220adantl 482 . . . . . . . . . . . 12 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ (𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂflds (0[,)+∞))))) → (𝑠(+g‘(ℂflds (0[,)+∞)))𝑡) = (𝑠(+g‘(ℝ*𝑠s (0[,)+∞)))𝑡))
222 funmpt 6539 . . . . . . . . . . . . 13 Fun (𝑦𝑥 ↦ (𝐹𝑦))
223222a1i 11 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → Fun (𝑦𝑥 ↦ (𝐹𝑦)))
224150, 177eleqtrdi 2848 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ (Base‘(ℂflds (0[,)+∞))))
225224ralrimiva 3143 . . . . . . . . . . . . 13 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → ∀𝑦𝑥 (𝐹𝑦) ∈ (Base‘(ℂflds (0[,)+∞))))
226151rnmptss 7070 . . . . . . . . . . . . 13 (∀𝑦𝑥 (𝐹𝑦) ∈ (Base‘(ℂflds (0[,)+∞))) → ran (𝑦𝑥 ↦ (𝐹𝑦)) ⊆ (Base‘(ℂflds (0[,)+∞))))
227225, 226syl 17 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → ran (𝑦𝑥 ↦ (𝐹𝑦)) ⊆ (Base‘(ℂflds (0[,)+∞))))
228169, 170, 171, 184, 193, 221, 223, 227gsumpropd2 18535 . . . . . . . . . . 11 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → ((ℂflds (0[,)+∞)) Σg (𝑦𝑥 ↦ (𝐹𝑦))) = ((ℝ*𝑠s (0[,)+∞)) Σg (𝑦𝑥 ↦ (𝐹𝑦))))
229165, 166, 2283eqtrd 2780 . . . . . . . . . 10 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (ℂfld Σg (𝑦𝑥 ↦ (𝐹𝑦))) = ((ℝ*𝑠s (0[,)+∞)) Σg (𝑦𝑥 ↦ (𝐹𝑦))))
23030adantl 482 . . . . . . . . . . 11 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ∈ Fin)
231148recnd 11183 . . . . . . . . . . 11 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ ℂ)
232230, 231gsumfsum 20864 . . . . . . . . . 10 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (ℂfld Σg (𝑦𝑥 ↦ (𝐹𝑦))) = Σ𝑦𝑥 (𝐹𝑦))
233229, 232eqtr3d 2778 . . . . . . . . 9 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → ((ℝ*𝑠s (0[,)+∞)) Σg (𝑦𝑥 ↦ (𝐹𝑦))) = Σ𝑦𝑥 (𝐹𝑦))
234101, 161, 2333eqtrrd 2781 . . . . . . . 8 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → Σ𝑦𝑥 (𝐹𝑦) = (𝐺 Σg (𝐹𝑥)))
235234mpteq2dva 5205 . . . . . . 7 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) = (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))))
236235rneqd 5893 . . . . . 6 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) = ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))))
237236supeq1d 9382 . . . . 5 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ*, < ) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ))
23898, 237eqtrd 2776 . . . 4 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ))
23993, 238pm2.61dan 811 . . 3 (𝜑 → (Σ^𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ))
24021, 9, 11, 1xrge0tsms 24197 . . 3 (𝜑 → (𝐺 tsums 𝐹) = {sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < )})
241239, 240eleq12d 2832 . 2 (𝜑 → ((Σ^𝐹) ∈ (𝐺 tsums 𝐹) ↔ sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) ∈ {sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < )}))
2428, 241mpbird 256 1 (𝜑 → (Σ^𝐹) ∈ (𝐺 tsums 𝐹))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2943  wral 3064  wrex 3073  Vcvv 3445  cin 3909  wss 3910  𝒫 cpw 4560  {csn 4586   class class class wbr 5105  cmpt 5188  dom cdm 5633  ran crn 5634  cres 5635  Fun wfun 6490   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  Fincfn 8883  supcsup 9376  cc 11049  cr 11050  0cc0 11051   + caddc 11054  +∞cpnf 11186  *cxr 11188   < clt 11189  cle 11190   +𝑒 cxad 13031  [,)cico 13266  [,]cicc 13267  Σcsu 15570  Basecbs 17083  s cress 17112  +gcplusg 17133   Σg cgsu 17322  *𝑠cxrs 17382  Mndcmnd 18556  SubMndcsubmnd 18600  CMndccmn 19562  SRingcsrg 19917  fldccnfld 20796   tsums ctsu 23477  Σ^csumge0 44593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129  ax-addf 11130  ax-mulf 11131
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-fi 9347  df-sup 9378  df-inf 9379  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-q 12874  df-rp 12916  df-xadd 13034  df-ioo 13268  df-ioc 13269  df-ico 13270  df-icc 13271  df-fz 13425  df-fzo 13568  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-sum 15571  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-starv 17148  df-tset 17152  df-ple 17153  df-ds 17155  df-unif 17156  df-rest 17304  df-topn 17305  df-0g 17323  df-gsum 17324  df-topgen 17325  df-ordt 17383  df-xrs 17384  df-mre 17466  df-mrc 17467  df-acs 17469  df-ps 18455  df-tsr 18456  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-submnd 18602  df-grp 18751  df-minusg 18752  df-mulg 18873  df-cntz 19097  df-cmn 19564  df-abl 19565  df-mgp 19897  df-ur 19914  df-srg 19918  df-ring 19966  df-cring 19967  df-fbas 20793  df-fg 20794  df-cnfld 20797  df-top 22243  df-topon 22260  df-topsp 22282  df-bases 22296  df-ntr 22371  df-nei 22449  df-cn 22578  df-haus 22666  df-fil 23197  df-fm 23289  df-flim 23290  df-flf 23291  df-tsms 23478  df-sumge0 44594
This theorem is referenced by: (None)
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